Calculating The Fine Structure Constant.

[IDMclean]

Something of note, I managed to come up with the indexing method for primes only because I implicitly neglect one prime number, the 0th: 2.

 

All primes greater than 2 are odd numbers, so they all follow the form of 2n+1. The reason for neglecting two besides it's unique status among the prime numbers comes from the way the natural number line is constructed:

the number zero, 0, is the successor to no number.

One, 1, is a number.

Any number + 1 is a number.

Analogously,

The prime, 2, is the successor to no prime. [math]P_0=\{2\}[/math]

The successor of any prime is of the form: [math]P_n = \{2n+1;i,j,n\in \mathbb{N}^+ \wedge n\not=2ij+i+j\}[/math]

 

I suspect if the nonfiguratives are countably infinite, the zeroth element will have a unique form with relation to the other nonfiguratives. It'll be like zero on the natural number line, you include it as an axiom and use the 1st element with a rule to generate the rest.

 

Btw, do we have a concise table of the first 10-20 nonfiguratives at hand for comparison?

[modest]

Non-figurate numbers:

7,8,11,13,14,17,19,20,23,26,29,31,32,37,38,41,43,44,47,50,53
,56,59,61,62,67,68,71,73,74,77,79,80,83,86,89,97,98,101,103,
104,107,109,110,113,116,119,122,127,128,131,134,137,139,140,143,
146,149,151,152,157,158,161,163,164,167,170,173,179,181,182,187,
188,191,193,194,197,199,200,203,206,209,211,212,218,221,223,224,
227,229,230,233,236,239,241,242,248,251,254,257,263,266,269,271,
272,277,278,281,283,284,290,293,296,299,302,307,308,311,313,314,
317,319,320,323,326,329,331,332,337,338,347,349,350,353,356,359,
362,367,368,371,373,374,377,379,380,383,386,389,391,392,397,398,
401,404,407,409,410,413,416,419,421,422,431,433,434,437,439,440,
443,446,449,452,457,458,461,463,464,467,470,473,476,479,482,487,
488,491,493,494,497,499,500,503,509,517,518,521,523,524,527,530,
533,536,539,541,542,547,548,551,554,557,563,566,569,571,572,577,
578,581,583,584,587,589,593,599,601,602,607,608,611,613,614,617,
619,620,623,626,629,631,632,638,641,643,644,647,649,650,653,656,
659,661,662,667,668,673,674,677,683,686,689,691,692,698,701,704,
707,709,710,713,716,719,722,727,728,731,733,734,737,739,740,743,
746,749,751,752,757,758,761,767,769,770,773,776,779,787,788,791,
794,797,799,800,803,806,809,811,812,817,818,821,823,824,827,829,
830,839,842,851,853,854,857,859,860,863,866,869,872,877,878,881,
883,884,887,890,893,896,899,901,902,907,908,911,913,914,917,919,
920,923,926,929,937,938,941,943,944,947,950,953,956,959,962,967,
968,971,974,977,979,980,983,986,989,991,992,997,998...

 

It is OEIS--A090467. There are lists of several millions of non-figs in 20236.

 

~modest

[IDMclean]

Perhaps obvious, but useful to explicitly catalog the properties of the numbers when doing analysis:

no nonfigurative number of the form [math]a^n[/math] (8 would invalidate this because it's of the form [math]2^3[/math]. Would restricting a to odd numbers fix the problem with this tentative rule?)

No nonfigurative number ends in 5 (no odd multiples of five).

Most even nonfiguratives are different from the previous and the successive even figurative by 6 and most are different from the previous nonfigurative by 1 with the exceptions occurring where there would be a number of the form a^n or 10n+5.

In the below sequence, the pattern of {n+1, n+3, n+2, n+1} seems to have some kind of recursive repetition.

Successive {n+3, n+3, n+3}, {n+3, n+3, n+3, n+3}, {n+3, n+3, n+3, n+3, n+3} patterns show up.

Also, all nonfiguratives so far follow some kind of recursive form: {n+1, n+2, n+3, n+5, n+6, n+8}. Can we say with confidence they all follow one of those six forms or will we eventually end up with an ever growing list of forms?

 

[math]\begin{array}{ccccccccccccccc} 7 & 8 & 11 & 13 & 14 & 17 & 19 & 20 & 23 & 26 & 29 & 31 & 32 & 37 & 38 \\ \hline

n & n+1 & n+3 & n+2 & n+1 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+5 & n+1 & n+3 & n+2 \end{array}[/math]

 

[math]\begin{array}{ccccccccccccccc} 41 & 43 & 44 & 47 & 50 & 53 & 56 & 59 & 61 & 62 & 67 & 68 & 71 & 73 & 74 \\ \hline

n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+3 & n+3 & n+2 & n+1 & n+5 & n+1 & n+3 & n+2 & n+1 \end{array}[/math]

 

[math]\begin{array}{ccccccccccccccc} 77 & 79 & 80 & 83 & 86 & 89 & 97 & 98 & 101 & 103 & 104 & 107 & 109 & 110 & 113 \\ \hline

n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+8 & n+1 & n+3 & n+2 & n+1 & n+3 & n+2 & n+1 & n+3 \\

\end{array}[/math]

 

[math]\begin{array}{ccccccccccccccc} 116 & 119 & 122 & 127 & 128 & 131 & 134 & 137 & 139 & 140 & 143 & 146 & 149 & 151 & 152 \\ \hline

n+3 & n+3 & n+3 & n+5 & n+1 & n+3 & n+3 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+2 & n+1 \\

\end{array}[/math]

 

[math]\begin{array}{ccccccccccccccc} 157 & 158 & 161 & 163 & 164 & 167 & 170 & 173 & 179 & 181 & 182 & 187 & 188 & 191 & 193 \\ \hline

n+5 & n+1 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+6 & n+2 & n+1 & n+5 & n+1 & n+3 & n+2 \\

\end{array}[/math]

 

[math]\begin{array}{ccccccccccccccc} 194 & 197 & 199 & 200 & 203 & 206 & 209 & 211 & 212 & 218 & 221 & 223 & 224 & 229 & 230 \\ \hline

n+1 & n+3 & n+2 & n+1 & n+3 & n+3 & n+3 & n+2 & n+1 & n+6 & n+3 & n+2 & n+1 & n+5 & n+1 \end{array}[/math]

 

Does anyone else see the patterns that I do?

What about the numbers 0-5? How do they fit into all this?

[modest]

IDMclean, I've answered you here, Re: Non-Figurate Numbers, where I think the thread might make the better context if that's alright (I ended up referencing a couple posts in that thread).

 

~modest

[Don Blazys]

By making adjustments to [math]k[/math], the function:

 

[math]B(x)-B(x)*\alpha*(\mu-k)^{-1}[/math]

 

can be made to give approximations that are almost exact

for regions of [math]\varpi(x)[/math] spanning billions.

 

Also, given recent determinations of [math]\varpi(x)[/math] we can now surmise the following:

 

1: Local fluctuations will eventually become inconsequential.

 

2: The value of [math]k[/math] rises and falls, but seems to be tending toward [math]0.[/math]

However, it may be tending towards some other value such as: [math]2*e+\alpha^{\frac{1}{2}}[/math] or [math]\frac{3*\Pi}{2}[/math]

in which case we may be able to determine a few extra digits of [math]\alpha[/math] by the time we get to [math]\varpi(10^{14})[/math].

 

3: The function: [math]B(x)-B(x)*\alpha*(\mu-k)^{-1}[/math] will always approximate [math]\varpi(x)[/math]

 

to a much higher degree of accuracy than [math]Li(x)[/math] approximates [math]\Pi(x)[/math].

 

All this is good news. :)

 

But with all of these various "heavy duty" computing options now open to us... (See Craig D's last post.)

 

what is our next step? :confused:

 

Don.

[Don Blazys]

A "little bird" tells me that we have a:

 

NASA Rocket Scientist / Excel Spreadsheet Wizard / Physicist / Raconteur / All Around Nice Person

 

among us.

 

I wonder if...

 

Don.

[modest]

Interesting stuff, Don.

 

~modest

[Don Blazys]

To Modest,

 

Quoting Modest:

Interesting stuff, Don.

 

Thanks Modest. :)

 

But if it weren't for Turtle, Donk and Yourself,

this interesting... Heck... DOWNRIGHT STRANGE correlation

would never have seen the light of day.

 

The data that's been trickling in on [math]\varpi(x)[/math] has pretty much stopped,

but this much, I'm reasonably sure of...

 

At [math]\varpi(10^{12})[/math], the random fluctuations are still only about [math]\pm900[/math] or so,

so at [math]x=10^{12}[/math], and [math]k=2*e[/math], we typically get calculations of [math]\alpha[/math]

that are good to about 9 or 10 decimal places!

 

Thus, we are on the right path to the most interesting "counting function" this side of [math]Li(x)[/math],

and I can't think of a more worthwhile challenge for any "coder" than to go after [math]\varpi(10^{18})[/math],

since that will probably allow us to determine the actual value of [math]k[/math]

and the actual value of [math]\alpha[/math] to 14 decimal places or so!

 

Since I know nothing about computers, all this is out of my hands,

and all I can do is offer moral support and cheer the rest of you Hypographers on.

 

:cheer: :cheer: :cheer:

Yeah... I do feel pretty silly jumping around like this, but it's all in the name of science!

 

So if i'm gonna be the cheerleader, why don't you be the project manager?

 

Don.

[Rade]

Let me offer a very simplistic and non mathematical suggestion. I am not a physicist, I just enjoy thinking about things in new ways.

 

As I understand the history of this problem with fine structure constant (reading Roger Penrose, The Road To Reality), there was time when the number 137.0359...was thought to be the exact number 137. So, let me take that approach to the problem. What would the number 137 mean for the atomic nucleus, in terms of a "coupling constant" for "fine structure" ? What does fine structure mean--structure of what ?

 

As explained by Penrose, 1/137 is a type of coupling constant related to the nuclear electromagnetic interaction within the "structure" of the atomic nucleus, whereas the weak force coupling constant ~ 10^-6, and strong force = 1. So, we can place 1/137 within the fine structure of the atomic nucleus. If this is not correct, someone please correct me.

 

OK, if this makes sense, then, what aspect of atomic nucleus would the number 137 be associated with ? Again, keeping in mind it is a coupling constant--that it is somehow related to how the atomic nucleus maintains its "structure" or identity as a thing that exists (I bring in a bit of philosophy, you do not have to agree it "really" exists--consider it the wavefunction if you wish).

 

Penrose gives that 1/sq.rt 137 = charge on the PROTON, e = 0.0854246. So, clearly the number 137 has something to do with PROTONS.

 

Penrose also explains that Sir Arthur Eddington spent a good part of his life trying to relate the number 137 (fine structure constant) to the total number of PROTONS in the universe.

 

Here is my suggestion. Eddington was close to the solution, however 137 is not related to the total number of protons in the universe, but it is related to the Z value (= number of protons) for the largest possible isotope of elements that can have any fine "structure" via the interactions of all the fundamental forces (strong, weak, electromagnetic, gravity). I believe I read that recently an unstable isotope of a new element with Z = 117 was produced. I suggest that the number 137 is related to "charge on the proton" that places a "limit" on how many PROTONS can coexist to form an isotope, stable or unstable. That is, laws of nature will never allow for a Z = 138 element to exist. OK, I know this is simplistic, but if anyone would have a comment to show where my logic is false it would be appreciated.

[Don Blazys]

Let [math]\alpha=137.035999084^{-1}[/math], [math]\mu=1836.1526724718[/math], [math]\pi=3.1415926535898[/math] and [math]e=2.718281828459[/math],

 

and let [math]\varpi(x)[/math] represent how many

"polygonal numbers of rank greater than 2"

there actually are under a given number [math]x[/math].

 

Then, for values of [math]k[/math] ranging from:

 

[math]k=e^{\frac{\pi}{2}}=4.8104773809654...[/math]

 

to

 

[math]k=2*e=5.436563656918...[/math],

 

the function:

 

[math]\varpi(x)\[/math] ~

 

[math]B(x)-B(x)*\alpha*(\mu-k)^-1=[/math]

 

[math]

\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)-[/math]

 

[math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\alpha*(\mu-k)^{-1}

[/math]

 

is not just "accurate", but "breathtakingly close"

and "often perfect" for values of [math]x[/math] well beyond [math] x=10^{12}[/math].

 

However, if we change the above value of [math]\alpha[/math] to [math]137^{-1}[/math],

then this function becomes much less accurate.

 

Don.

[modest]

What does fine structure mean--structure of what ?

 

'Fine structure' refers to the spectral lines of an element (typically hydrogen). A spectral line, if you look closely enough (at its fine structure), is not a line. There's a fair Hyperphysics article:

 

Hydrogen Fine Structure

 

It refers, essentially, to the size of the 'splitting' of the spectral lines of hydrogen.

 

OK, if this makes sense, then, what aspect of atomic nucleus would the number 137 be associated with ?

 

A coupling constant determines the strength of an interaction. The fine structure constant determines the strength of the electromagnetic force on an electron. If it were different then the strength of that interaction would be different. I would presume there would be many other consequences because many parameters rely on the fine structure constant (as, for example, in the standard model).

 

Penrose also explains that Sir Arthur Eddington spent a good part of his life trying to relate the number 137 (fine structure constant) to the total number of PROTONS in the universe.

 

In fact, Eddington made many 'proofs' that the fine structure constant was exactly 1/136. When experimental measurement showed the value closer to 1/137 he made many more 'proofs' that it must be exactly 1/137. That is somewhat telling.

 

Certainly, if you asked most physicists they would tell you that the fine structure constant is one of those dimensionless parameters that can't be mathematically derived (though many have tried and as this thread attests, still try) because it is a fundamental physical constant and not a mathematical constant.

 

But, conventional wisdom is no reason not to try :naughty:

 

~modest

[Pyrotex]

I'm just curious, but why did you not simplify your equation to this:

 

[math]\varpi(x)\[/math] ~

 

[math]B(x)-B(x)*\alpha*(\mu-k)^-1=[/math]

 

[math]B(x)*(1-\alpha*(\mu-k)^-1)=[/math]

 

[math]

\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\left(1-\frac{\alpha}{(\mu-k)}\right)

[/math]

 

or even to:

 

[math]

\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\left(\frac{\mu-k-\alpha}{\mu-k}\right)

[/math]

[Don Blazys]

To: Modest,

 

Quoting Modest:

In fact, Eddington made many 'proofs' that the fine structure constant was exactly 1/136. When experimental measurement showed the value closer to 1/137 he made many more 'proofs' that it must be exactly 1/137. That is somewhat telling.

 

Certainly, if you asked most physicists they would tell you that the fine structure constant is one of those dimensionless parameters that can't be mathematically derived (though many have tried and as this thread attests, still try) because it is a fundamental physical constant and not a mathematical constant.

 

But, conventional wisdom is no reason not to try

 

Feynman suggested that it might be possible to derive the fine structure constant

from some purely mathematical relationship, possibly involving [math]\pi[/math] or [math]e[/math].

 

Eddington, Gilson, and a host of others tried very hard to do just that.

 

We, on the other hand, wanted nothing more than to provide Turtle with a

"counting function" for his "non-figurate numbers"!

 

The fine structure constant couldn't have been further from our minds.

 

Little indeed did we know that such an innocent search for a "counting function"

would result in this incredibly wierd... THING (as Capt. Kirk might say !).

 

In short, this relationship between "Turtles numbers" and the fine structure constant

was totally unexpected and came as a complete surprise

and even as somewhat of a shock :eek::eek2: ...and in that sense,

was a completely accidental and serendipidous discovery.

 

But even if the fine structure constant were not involved,

this function would still be important,

because it's the only counting function for

"non-trivial polygonal numbers" that we have...

and considering how extraordinarily difficult it is

to calculate them by computer,

this is something that we as Hypographers should be

proud to contribute to the annals of mathematical literature.

 

Don.

[Don Blazys]

To: Pyrotex,

 

Quoting Pyrotex:

I'm just curious, but why did you not simplify your equation to this:

 

I wanted to show that we are essentially subtracting an "error term".

 

Also, (and please keep this a secret) I tend to make really stupid blunders when simplifying,

which I can't afford to do, since I already have a reputation as a "crank" and a "crackpot".

 

Anyway, thanks for simplifying it for me.

 

It looks beautiful !

 

Don.

[Don Blazys]

To: Pyrotex,

 

I just remembered...:doh:

 

You are the N.A.S.A. scientist! :)

 

Wow! ...and Please...

 

If you can find some spare time, can you and your colleagues

use one of your more powerful computers to calculate [math]\varpi(x)[/math]

to some really impressive value of [math]x[/math] as a "side project"?

 

Don.

 

P.S.

I live in LaCrescenta,

just a "stone throw" away from N.A.S.A./J.P.L in Pasadena,

and my son used to play baseball for P.C.C.

[modest]

Let [math]\alpha=137.035999084^{-1}[/math], [math]\mu=1836.1526724718[/math], [math]\pi=3.1415926535898[/math] and [math]e=2.718281828459[/math],

 

and let [math]\varpi(x)[/math] represent how many

"polygonal numbers of rank greater than 2"

there actually are under a given number [math]x[/math].

 

Then, for values of [math]k[/math] ranging from:

 

[math]k=e^{\frac{\pi}{2}}=4.8104773809654...[/math]

 

to

 

[math]k=2*e=5.436563656918...[/math],

 

the function:

 

[math]\varpi(x)\[/math] ~

 

[math]B(x)-B(x)*\alpha*(\mu-k)^-1=[/math]

 

[math]

\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)-[/math]

 

[math]\left(x-(\alpha*\Pi*e+e)^{-1}*x-\frac{1}{2}*\sqrt{x-(\alpha*\Pi*e+e)^{-1}*x}\right)*\alpha*(\mu-k)^{-1}

[/math]

 

is not just "accurate", but "breathtakingly close"

and "often perfect" for values of [math]x[/math] well beyond [math] x=10^{12}[/math].

 

Here is output I get for K = 4.8104773809654... and k = 5.436563656918...

 

x = 1000   w(x,k) between:
627.71007436564630089122291274484991368745775233745
627.71007351024513457965178477714116184532220891766

x = 2000   w(x,k) between:
1262.8319109844068048188315028082273223220408149266
1262.8319092635042199699382106237052357330017876901

x = 3000   w(x,k) between:
1899.1731653019497577233789712653718430130891941325
1899.1731627138840137273547424210038181160456364800

x = 4000   w(x,k) between:
2536.1456366682526841060733678990588046556714731561
2536.1456332121636007272406283339013719508493169182

x = 5000   w(x,k) between:
3173.5214905136186968352994675234152455860152810070
3173.5214861889565715005253535614115241913733150994

x = 6000   w(x,k) between:
3811.1838793999144923299121008212487022788435608051
3811.1838742062888543136022104660349911590603128445

x = 7000   w(x,k) between:
4449.0633905891958270956782645636886500015399292956
4449.0633845263107966935060789212318882849924672414

x = 8000   w(x,k) between:
5087.1147978427337742849180904351725992055935668155
5087.1147909103551030757505388066485679864133735458

x = 9000   w(x,k) between:
5725.3066869078191496445513654626266729046411624560
5725.3066791057553984427665306702806303165813240016

x = 10000   w(x,k) between:
6363.6161852582677540104780344619532545008406233396
6363.6161765863586527742569376131926679173167096429

x = 11000   w(x,k) between:
7002.0260259843458586823403269283054852015736776984
7002.0260164424546675557380243860749699782828475886

x = 12000   w(x,k) between:
7640.5227944147797624176405697588206178684462937088
7640.5227840027880221585206611808647810127059702088

x = 13000   w(x,k) between:
8279.0958225657188960328414256860179558988657522187
8279.0958112835226850115746731703100369784604159804

x = 14000   w(x,k) between:
8917.7364608768676008081803480143852305588163358367
8917.7364487243747844354721233583182160727156697147

x = 15000   w(x,k) between:
9556.4375808703607296340964922209375570900127286821
9556.4375678474888875319240434969877222284610081661

x = 16000   w(x,k) between:
10195.193225084345697506656905435553518434366820237
10195.193211191020527726229491786278936942518230168

x = 17000   w(x,k) between:
10833.998354245635334437776992402841346714329201774
10833.998339481789402213638510477183303282603609887

x = 18000   w(x,k) between:
11472.848660574980908398259762880835830650658255667
11472.848644940552649317436984548829996760234757567

x = 19000   w(x,k) between:
12111.740427236359597984801623601750946253438883831
12111.740410731292512689031687340149415458002271113

x = 20000   w(x,k) between:
12750.670420712037805288736647093625377575810404421
12750.670403336279800840736001020918289683771125614

x = 21000   w(x,k) between:
13389.635807139828404914441557531301911421554997988
13389.635788893331250174146461903922396810761356316

x = 22000   w(x,k) between:
14028.634086397590677042683204910369707789178739808
14028.634067280309547868710408516885120554138463198

x = 23000   w(x,k) between:
14667.663039539710267965840542879606793788184838634
14667.663019551603364046985748200036589297529503939

x = 24000   w(x,k) between:
15306.720686421589478601589456098651268242222932498
15306.720665562617698034826773609784271879350020186

x = 25000   w(x,k) between:
15945.805251197832327692389987817839341244848446500
15945.805229467958988577629511681896291828660035417

x = 26000   w(x,k) between:
16584.915133976589557872722287424708600097606688996
16584.915111375780158535124136922210451180801621841

x = 27000   w(x,k) between:
17224.048887338490014082784795480346324566071298729
17224.048863866712025293497756279582888689981001186

x = 28000   w(x,k) between:
17863.205196737112230456749325992890140968371637960
17863.205172394334914362571569299797114391419228063

x = 29000   w(x,k) between:
18502.382864024440433047570352757287385735075082746
18502.382838810634684551013175257983771426637438465

x = 30000   w(x,k) between:
19141.580793513081528904407546795923314703672441691
19141.580767428219736037234490756809065500662775883

x = 31000   w(x,k) between:
19780.797980113560889582780220311356412396031180985
19780.797953157616810045243442762517041807502904840

x = 32000   w(x,k) between:
20420.033499181148207158856520145217016657234881482
20420.033471354096858694997547850249522054465939755

x = 33000   w(x,k) between:
21059.286497780415499363007066151517335515757133173
21059.286469082233062040031701700753686478926112857

x = 34000   w(x,k) between:
21698.556187132826466330671940799134407131778292698
21698.556157563490195110243582196464009567402727527

x = 35000   w(x,k) between:
22337.841836057237701464620238345688296505852242667
22337.841805616725847712997967524971076774784129070

x = 36000   w(x,k) between:
22977.142765248279040201750612609484141336086041244
22977.142733936570780996966590357132694975006739749

x = 37000   w(x,k) between:
23616.458342265401190655434563372087604321885042478
23616.458310082476564923061048865777091038824893879

x = 38000   w(x,k) between:
24255.787977127593337868919399947541112601679049441
24255.787944073433188521058998887575103444331868081

x = 39000   w(x,k) between:
24895.131118426628641325102703738359449723198731757
24895.131084501214562695986828410017326160947337514

x = 40000   w(x,k) between:
25534.487249886136331257231493017496281809240931491
25534.487215089450621227884463022575818436682410114

x = 41000   w(x,k) between:
26173.855887305547110008225371993243664182832132534
26173.855851637572726295714642790522382053451455487

x = 42000   w(x,k) between:
26813.236575837569091350450836894722109527550019341
26813.236539298289611507210320875500680602416945063

x = 43000   w(x,k) between:
27452.628887555754907466659201578062926564284359228
27452.628850145154492194568905209859546724654319131

x = 44000   w(x,k) between:
28092.032419275252622924583620359559720491463008690
28092.032380993315982358783695347959470818819585089

x = 45000   w(x,k) between:
28731.446790595257325360311538710630395969385370240
28731.446751441969688020631942378520296477193431542

x = 46000   w(x,k) between:
29370.871642136204938666824901315767871184044443313
29370.871602111552022811449033717200655410878791870

x = 47000   w(x,k) between:
30010.306633948540351274994625581397958686322602729
30010.306593052508338422546980386669342389273590420

x = 48000   w(x,k) between:
30649.751444073080512718811648429948963105964209192
30649.751402305656023132286027717158141147870729411

x = 49000   w(x,k) between:
31289.205767235685835034738779810487918708079604468
31289.205724596855904984240727811988146381558343164

x = 50000   w(x,k) between:
31928.669313661235670566708362228316157225323444258
31928.669270150987731208432228710907143869106920108

 

x = 10   w(x,k) between:
5.1383604794014263845590518660638373911117566520885
5.1383604723992135374080108326312590128230004330355

x = 100   w(x,k) between:
60.035148954194738355415409361983785673743876517396
60.035148872382866071854017310490695074639466376989

x = 1000   w(x,k) between:
627.71007436564630089122291274484991368745775233745
627.71007351024513457965178477714116184532220891766

x = 10000   w(x,k) between:
6363.6161852582677540104780344619532545008406233396
6363.6161765863586527742569376131926679173167096429

x = 100000   w(x,k) between:
63909.747700819666713475468535869653114508596105030
63909.747613727751266353685492854268790162442547906

x = 1000000   w(x,k) between:
639962.63142421471458573717442883107232134859204996
639962.63055211658573331424582696049839626537168373

x = 10000000   w(x,k) between:
6402362.1727245170375910786262008119289084878192159
6402362.1639998075047192536194368284050725464713521

x = 100000000   w(x,k) between:
64032273.271405350850420611152709464700847504502156
64032273.184146465777890310103352462155671874175567

x = 1000000000   w(x,k) between:
640350091.29887720742154318034621965906542506400872
640350090.42625107425276421264519685576781766930082

x = 10000000000   w(x,k) between:
6403587428.4303738789605300523692100447718769031872
6403587419.7039946498346198655418103387276407876287

x = 100000000000   w(x,k) between:
64036147870.151975778778671211883350568288269221743
64036147782.887810663084809771534825729387397151738

x = 1000000000000   w(x,k) between:
640362343855.93577583523769460790364022405895484844
640362342983.29294570391787510624667510436861246359

 

Is this the right post from Donk for a comparison?

 

Ok, Don - here are your figures:

  NO           FIG     NONFIG    PRIME*  NONFIG/NONPRIME
1000    :      627       373      165       208 
2000    :     1263       737      300       437 
3000    :     1901      1099      427       672 
4000    :     2540      1460      547       913 
5000    :     3174      1826      666      1160 
6000    :     3814      2186      780      1406 
7000    :     4451      2549      897      1652 
8000    :     5089      2911     1004      1907 
9000    :     5723      3277     1114      2163 
10000    :     6362      3638     1226      2412 
11000    :     7000      4000     1332      2668 
12000    :     7638      4362     1435      2927 
13000    :     8273      4727     1544      3183 
14000    :     8914      5086     1649      3437 
15000    :     9555      5445     1751      3694 
16000    :    10190      5810     1859      3951 
17000    :    10833      6167     1957      4210 
18000    :    11474      6526     2061      4465 
19000    :    12112      6888     2155      4733 
20000    :    12751      7249     2259      4990 
21000    :    13390      7610     2357      5253 
22000    :    14030      7970     2461      5509 
23000    :    14664      8336     2561      5775 
24000    :    15305      8695     2665      6030 
25000    :    15942      9058     2759      6299 
26000    :    16580      9420     2857      6563 
27000    :    17216      9784     2958      6826 
28000    :    17857     10143     3052      7091 
29000    :    18495     10505     3150      7355 
30000    :    19137     10863     3242      7621 
31000    :    19777     11223     3337      7886 
32000    :    20414     11586     3429      8157 
33000    :    21055     11945     3535      8410 
34000    :    21693     12307     3635      8672 
35000    :    22338     12662     3729      8933 
36000    :    22976     13024     3821      9203 
37000    :    23614     13386     3920      9466 
38000    :    24254     13746     4014      9732 
39000    :    24891     14109     4104     10005 
40000    :    25530     14470     4200     10270 
41000    :    26173     14827     4288     10539 
42000    :    26813     15187     4389     10798 
43000    :    27454     15546     4491     11055 
44000    :    28090     15910     4576     11334 
45000    :    28733     16267     4672     11595 
46000    :    29369     16631     4758     11873 
47000    :    30009     16991     4848     12143 
48000    :    30648     17352     4943     12409 
49000    :    31287     17713     5032     12681 
50000    :    31924     18076     5130     12946 

[font="Courier New"]         NO        FIG      NONFIG      PRIME*    NONFIG/NONPRIME
          10 :           8           2          1           1 
         100 :          62          38         22          16 
       1,000 :         627         373        165         208 
      10,000 :       6,362       3,638      1,226       2,412 
     100,000 :      63,894      36,106      9,589      26,517 
   1,000,000 :     639,951     360,049     78,495     281,554 
  10,000,000 :   6,402,330   3,597,670    664,576   2,933,094 
 100,000,000 :  64,032,126  35,967,874  5,761,452  30,206,422
1,000,000,000 : 640,349,984 359,650,016 50,847,531 308,802,485[/font]

* Only nonfigurate primes - 2, 3 and 5 have been left out of the count.

 

I now have a massive amount of raw data, with figurates running up to a billion, and can pull it out in just about any form you need. Would powers of 2 be better than powers of 10? You'd get 29 data points instead of 8. Or powers of e, for 20 data points?

 

I've checked the data as best I can - one check is that my primes list matches that on wikipedia, remembering to subtract 3 from the powers-of-ten value. If anyone comes up with an alternative nonfig-generator, I'd love to send a million or so nonfigs to them as a crosscheck :)

 

~modest

[Don Blazys]

[math]x[/math]______[math]\varpi(x)[/math]_____[math]B(x)*\left(\frac{\mu-2*e-\alpha}{\mu-2*e}\right)[/math] ____Diff____%Error___

10^1____3_______________5_____________________2_____.66666

10^2____57______________60____________________3_____.05263

10^3____622_____________628___________________6_____.00964

10^4____6,357___________6,364__________________7_____.00110

10^5____63,889__________63,910_________________21____.00032

10^6____639,946_________639,963________________17____.00002656

10^7____6,402,325________6,402,362______________37____.00000577915

10^8____64,032,121_______64,032,273_____________152___.00000238943

10^9____640,349,979______640,350,090____________111___.00000017334

10^10___6,403,587,409____6,403,587,420___________11____.00000001343

10^11___64,036,148,166___64,036,147,783__________-383__-.00000000598

10^12___640,362,343,980__640,362,342,983_________-997__-.00000000156

 

[math]B(x)*\left(\frac{\mu-2*e-\alpha}{\mu-2*e}\right)[/math] crosses [math]\varpi(x)[/math] many, many times

 

between [math]x=[/math]10^11, and [math]x=[/math]10^12 .

 

Don.